Problem: Hungry Harry is a giant ogre with an even bigger appetite. After Harry wakes up from hibernation, his daily hunger $H(t)$ (in $\text{kg}$ of pigs) as a function of time $t$ (in hours) can be modeled by a sinusoidal expression of the form $a\cdot\cos(b\cdot t)+d$. When Harry wakes up at $t=0$, his hunger is at a maximum, and he desires $30 \text{ kg}$ of pigs. Within $2$ hours, his hunger subsides to its minimum, when he only desires $15 \text{ kg}$ of pigs. Find $H(t)$. $\textit{t}$ should be in radians. $H(t) = $
The strategy First, we should convert the given information about the real-world context into mathematical terms of the sinusoidal function and its graph. Then, we should use the given information to find the amplitude, midline, and period of the function's graph. Finally, we should find $a$, $b$, and $d$ in the expression $a\cos(b\cdot t)+d$ by considering the features we found. Converting the given information into mathematical terms At $t=0$, Harry wants $30\text{ kg}$ of pigs. This means the graph of the function passes through $(0,30)$. We are given that this is the point at which Harry's hunger is at a maximum, which corresponds to a maximum point of the graph. $2$ hours later (which means $t=2$ ), Harry wants $15\text{ kg}$ of pigs. This corresponds to the point $(2,15)$. We are given that this is the point at which Harry's hunger is at a minimum, which corresponds to a minimum point of the graph. In conclusion, the graph has a maximum at $(0,30)$ and then has a minimum point at $(2,15)$. Determining the amplitude, midline, and period The midline intersection is halfway between the maximum and minimum, which is at $y={22.5}$, so this is the midline. The minimum point is $7.5$ units below the midline, so the amplitude is ${7.5}$. The minimum point is $2$ units to the right of the nearest maximum, so the period is $2\cdot 2={4}$. [Why did we multiply by 2?] Determining the parameters in $a\cos(b\cdot t)+d$ Since the maximum at $t=0$ is followed by a minimum point, we know that $a>0$. [How do we know that?] The amplitude is ${7.5}$, so $|a|={7.5}$. Since $a>0$, we can conclude that $a=7.5$. The midline is $y={22.5}$, so $d=22.5$. The period is ${4}$, so $b=\dfrac{2\pi}{{4}}=\dfrac{\pi}{2}$. The answer $H(t)=7.5\cos\left(\dfrac{\pi}{2}t\right)+22.5$